永久序列的渐近性质
Asymptotic properties of permanental sequences
摘要：
设$U=\{U_{j,k},j,k\in \overline {\mathbb N}\}$是瞬态对称Borel右过程$X$的势，其状态空间为$\overline {\mathbb N}$ .对于$X$的任何过量函数$f=\{f_{k,k\in \overline {\mathbb N}}\}$，$\widetilde U=\{\widetilde U_{j,k},j,k\in\overline {\mathbb N}\}$，其中\begin{equation} \widetilde U_{j,k}= U_{j,k} +f_{ k},\qquad j,k\in\overline {\mathbb N},\label{a.1} \end{equation}是$α$的内核-所有$α>0$的永久序列$\widetilde X_α=(\widetilde X_{α, 1} ,\ldots)$。对称势$U$也是平均零高斯序列$η=\{η_{j},j\in \overline {\mathbb N}\}$的协方差。条件给出的潜力$U$和过量的功能$f$下，\begin{equation} \limsup_{j\to \infty}\frac{ η_{j}}{( 2\,ϕ_{j})^{1/2} }=1 \quad a.s. \quad \implies \quad \limsup_{n\to \infty}\frac{\widetilde X_{α, j}}{ϕ_{j} }=1\quad a.s.,\label{a.2} \end{equation}的所有$α>0$，和序列$ϕ=\{ϕ_{j}\}$，使$f_{j}=o(ϕ_{j})$。函数$ϕ$由$U$确定。给出了$U$是有迁移和无迁移的对称生灭过程、一阶和高阶高斯自回归序列以及Lévy过程在$\mathbf Z$上的势的许多例子。
英文摘要：
Let $U=\{U_{j,k},j,k\in \overline {\mathbb N}\}$ be the potential of a transient symmetric Borel right process $X$ with state space $\overline {\mathbb N}$. For any excessive function $f=\{f_{k,k\in \overline {\mathbb N}}\}$ for $X$ , $\widetilde U=\{\widetilde U_{j,k},j,k\in\overline {\mathbb N}\}$, where \begin{equation} \widetilde U_{j,k}= U_{j,k} +f_{ k},\qquad j,k\in\overline {\mathbb N},\label{a.1} \end{equation} is the kernel of an $α$-permanental sequence $\widetilde X_α=(\widetilde X_{α, 1} ,\ldots)$ for all $α>0$. The symmetric potential $U$ is also the covariance of a mean zero Gaussian sequence $η=\{η_{j},j\in \overline {\mathbb N}\}$. Conditions are given on the potentials $U$ and excessive functions $f$ under which, \begin{equation} \limsup_{j\to \infty}\frac{ η_{j}}{( 2\,ϕ_{j})^{1/2} }=1 \quad a.s. \quad \implies \quad \limsup_{n\to \infty}\frac{\widetilde X_{α, j}}{ϕ_{j} }=1\quad a.s.,\label{a.2} \end{equation} for all $α>0$, and sequences $ϕ=\{ϕ_{j}\}$ such that $f_{j}=o(ϕ_{j})$. The function $ϕ$ is determined by $U$. Many examples are given in which $U$ is the potential of symmetric birth and death processes with and without emigration, first and higher order Gaussian autoregressive sequences and Lévy processes on $\mathbf Z$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
